Metamath Proof Explorer

Theorem chslej

Description: Subspace sum is smaller than subspace join. Remark in Kalmbach p. 65. (Contributed by NM, 12-Jul-2004) (New usage is discouraged.)

Ref Expression
Assertion chslej 
|- ( ( A e. CH /\ B e. CH ) -> ( A +H B ) C_ ( A vH B ) )

Proof

Step Hyp Ref Expression
1 chsh 
 |-  ( A e. CH -> A e. SH )
2 chsh 
 |-  ( B e. CH -> B e. SH )
3 shslej 
 |-  ( ( A e. SH /\ B e. SH ) -> ( A +H B ) C_ ( A vH B ) )
4 1 2 3 syl2an 
 |-  ( ( A e. CH /\ B e. CH ) -> ( A +H B ) C_ ( A vH B ) )